PHYSICAL REVIEW B
VOLUME 62, NUMBER 18
1 NOVEMBER 2000-II
Interaction between a vortex and a columnar defect in the London approximation
H. Nordborg* and V. M. Vinokur
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60469
共Received 20 January 2000兲
We calculate the interaction between a vortex and an insulating cylindrical cavity in a type-II superconductor
using the London approximation, thus extending the result in an earlier work by Mkrtchyan and Shmidt „Zh.
Éksp. Teor. Fiz. 61, 367 共1971兲 关Sov. Phys. JETP 34, 195 共1972兲兴… to an arbitrarily large cavity radius. In the
limit of an infinitely large radius our result reduces to the well-known Bean-Livingston formula for the
interaction between a vortex and an insulating wall.
I. INTRODUCTION
The interaction between a vortex and an insulating cylindrical cavity in a type-II superconductor was calculated in
1972 by Mkrtchyan and Shmidt1 and was long regarded as a
somewhat academic exercise in the phenomenology of the
vortex state. Their result has gained importance in recent
years in connection with the extensive study of high-T c superconductors containing artificially manufactured columnar
defects due to irradiation with heavy ions 共see, for example
Ref. 2 for a recent review兲. Modeling an amorphous track
left by a heavy ion as an insulating cylindrical cavity, the
Mkrtchyan-Shmidt maximal pinning force gives a reasonable
estimate for a low-field–temperature columnar defect critical
current. Making use of the full interaction between a vortex
and a cavity one derives a temperature dependence of the
critical current,3,4 which agrees excellently with experiments.
The work by Mkrtchyan and Shmidt, which henceforth will
be referred to as Ref. 1, has thus become the basic reference
to researchers working in the field of high-temperature superconductivity 共HTS兲.
Interestingly, Ref. 1, which was completed long before
the discovery of HTS, seems to be designed specifically for a
heavy ion tracks as it only considers a cylinder of radius b
Ⰶ␭, where ␭ is the London penetration depth. Recent developments in artificially engineered pinning structures 共see, for
example Ref. 5兲 raise the question of the interaction between
a vortex and a large cavity, with the radius b comparable to
or even exceeding ␭. The results of Ref. 1 could have been
easily extended to arbitrary b if not for an unfortunate oversight in the calculations, which invalidates the result at large
values of b. One can easily see, in particular, that taking the
limit of an infinitely large cavity in Ref. 1 does not reproduce
the Bean-Livingston barrier for a vortex interacting with a
flat interface. Motivated by the quest for a general result
valid in the whole range of parameters we present more general and somewhat more transparent derivation for the interaction of a vortex with a cylindrical cavity. We also demonstrate that the result of Ref. 1 for small cavities can very
easily and instructively be obtained using the method of image vortices similar to Ref. 6, as has already been pointed out
by Buzdin and Feinberg.7,8 A very similar problem has been
discussed by Wei et al., who considered the interaction between a vortex and a cylindrical superconducting column
with a different penetration depth.9 There is no a simple way,
0163-1829/2000/62共18兲/12408共5兲/$15.00
PRB 62
however, to generalize their result into the result presented
here, since the boundary conditions at the edge of the cavity
are completely different.
We consider a superconductor with an infinitely long cylindrical cavity, i.e., a columnar defect, of radius b. The induction is taken to be parallel with the column, which is
directed along the z axis. We thus have a two-dimensional
problem and only have to solve the London equation for the
z component B(x,y) of the induction. We measure all
lengths in units of the London penetration depth ␭ and measure the induction in units of 冑2H c ⫽⌽ 0 /2␲ ␭ ␰ , where H c is
the thermodynamic critical field, ⌽ 0 is the flux quantum, and
␰ is the superconducting coherence length. The London
equation for the magnetic induction from a vortex is then
given by
B⫺⌬B⫽
2 ␲ (2)
␦ 共 r⫺r0 兲 ,
␬
共1兲
where ␬ ⫽␭/ ␰ is the Ginzburg parameter and r0 is the position of the vortex. The boundary condition for the cavity is
Js •n⫽0,
共2兲
i.e., the supercurrent is tangential to the boundary. Since the
current is given by the curl of the magnetic induction, it
follows that B(r) is constant on the boundary of the cavity.
From the fact that B is a harmonic function inside the cavity,
it then follows that the induction is constant, B(r)⫽B 0 , inside the entire cavity.
II. SOLVING THE LONDON EQUATION
We take the center of the column to be the origin of our
coordinate system and use polar coordinates (r, ␾ ). It is convenient to split the induction in two parts
B 共 r, ␾ 兲 ⫽B 1 共 r 兲 ⫹B 2 共 r, ␾ 兲 ,
共3兲
where B 1 is the radially symmetric field due to the columnar
defect and B 2 is the contribution from the vortex. The field
B 1 satisfies the homogeneous London equation
⳵ 2B 1
⳵r
12 408
2
⫹
1 ⳵B1
⫺B 1 ⫽0,
r ⳵r
共4兲
©2000 The American Physical Society
PRB 62
INTERACTION BETWEEN A VORTEX AND A COLUMNAR . . .
with the boundary condition B 1 (b)⫽B 0 . The solution is easily found to be
K 0共 r 兲
,
B 1 共 r 兲 ⫽B 0
K 0共 b 兲
共5兲
r⭓b.
The field B 2 is the solution to the inhomogeneous equation
1 ⳵ B 2 1 ⳵ 2B 2
2␲
⫹
⫺B
⫽⫺
␦ 共 ␾ 兲 ␦ 共 r⫺r 0 兲 ,
⫹
2
r ⳵r
␬r
⳵r2
r2 ⳵␾2
A comparison of Eqs. 共10兲 and 共12兲 shows that we can write
B 2 共 r兲 ⫽
B c2 共 r, ␾ 兲 ⬅B c2 共 r, ␾ ;r 0 兲 ⫽
共6兲
with the boundary condition B 2 (b, ␾ )⫽0. This equation is
solved using separation of variables
兺n B̃ n共 r 兲 e in ␾ ,
共7兲
and the solution has the form
B̃ n 共 r 兲 ⫽ ␣ n I n 共 r 兲 ⫹ ␤ n K n 共 r 兲 ,
B̃ n 共 r 兲 ⫽ ␥ n K n 共 r 兲 ,
共8a兲
b⭐r⭐r 0 ,
r 0 ⭐r.
共8b兲
In order to find the constants ␣ n , ␤ n , and ␥ n we use the fact
that B̃ n (b)⫽0, B̃ n (r) should be continuous at r⫽r 0 , and the
discontinuity in the derivate of B̃ n (r) due to the ␦ function is
given by
dB̃ n
dr
冏
⫺
r0⫹
冏
dB̃ n
dr
⫽⫺
r0⫺
1
.
␬r0
再
冎
1
I n共 b 兲 K n共 r 兲
K n共 r 0 兲 I n共 r 兲 ⫺
,
␬
K n共 b 兲
B̃ n 共 r 兲 ⫽
再
冎
兺n B̃ nv共 r 兲 e in ␾
兺n B̃ cn共 r 兲 e in ␾ ,
共15兲
1 I n共 b 兲 K n共 r 0 兲
K n共 r 兲 ,
␬
K n共 b 兲
r 0 ⬎b.
共16兲
The advantage of writing the field B 2 (r) in the form of Eq.
共14兲 is that the divergence at r⫽r0 is taken care of explicitly
and that the term B c2 (r, ␾ ;r 0 ) is well defined for all r⬎b. It
follows directly from Eq. 共16兲, or from the fact that
B 2 (b,0)⫽0, that for r0 ⫽(b,0) we have
B c2 共 r, ␾ ;b 兲 ⫽⫺
1
K 关 兩 r⫺ 共 b,0兲 兩 兴 .
␬ 0
共17兲
This property will be useful when computing the energy.
In order to find the induction inside the cavity we use the
fact that the order parameter has to be single valued. This
leads to the usual condition for flux quantization 共see, e.g.,
Ref. 10兲,
“⫻B⫽ ␬ ⫺1 ⵜ ␪ ⫺A,
共18兲
where ␪ is the phase of the order parameter and A is the
vector potential. Integrating this along the perimeter of the
cavity we obtain
共10a兲
r 0 ⭐r.
共10b兲
In order to better understand the solution for the field B 2 ,
and to obtain a form which can be implemented numerically,
we compare it to the solution for a vortex at r0 without the
cavity. We call this solution B v (r, ␾ ) and find
B v 共 r, ␾ 兲 ⫽
B̃ cn 共 r 兲 ⫽⫺
b⭐r⭐r 0 ,
1
I n共 b 兲 K n共 r 0 兲
K 共 r 兲 I n共 r 0 兲 ⫺
,
␬ n
K n共 b 兲
共14兲
where
共9兲
Solving for the constants, we find the solution
B̃ n 共 r 兲 ⫽
1
K 共 兩 r⫺r0 兩 兲 ⫹B c2 共 r, ␾ 兲 ,
␬ 0
where B c2 , which represents the modification of the vortex
field due to the column, is given by
⳵ 2B 2
B 2 共 r, ␾ 兲 ⫽
12 409
共11兲
with
b
冕
d ␾ 共 “⫻B兲 ␾ ⫽
2␲q
⫺ ␲ b 2B 0 ,
␬
共19兲
where q is the number of flux quanta in the cavity. The
integral over ␾ can easily be computed using
共 “⫻B兲 ␾ ⫽⫺
⳵B
.
⳵r
共20兲
and the fact that only the terms with n⫽0 survive. From this
we obtain
B 0⫽
1 K 0 共 r 0 兲 ⫹qK 0 共 b 兲
,
␬ b K 1 共 b 兲 ⫹ 21 bK 0 共 b 兲
共21兲
which is identical to Eq. 共8兲 in Ref. 1.
B̃ vn 共 r 兲 ⫽
1
K 共 r 兲I 共 r 兲,
␬ n 0 n
B̃ nv 共 r 兲 ⫽
1
I 共 r 兲K 共 r 兲,
␬ n 0 n
0⭐r⭐r 0 ,
r 0 ⭐r.
共12a兲
共12b兲
On the other hand, we know that the field from the vortex at
r0 is given by
B v 共 r兲 ⫽
1
K 共 兩 r⫺r0 兩 兲 .
␬ 0
共13兲
III. COMPUTING THE INTERACTION ENERGY
The next task is to compute the energy of the system,
which in the London approximation is given by
F⫽
冕
d 2 r 兵 B2 ⫹ 共 “⫻B兲 2 其 ,
共22兲
where the energy is measured in units of B 2c ␭ 2 /4␲ , lengths in
units of ␭, and the induction in units of 冑2B c . Here, and in
the following, we always refer to the energy per unit length
12 410
H. NORDBORG AND V. M. VINOKUR
PRB 62
of the vortex. The integral 共22兲 can be evaluated directly
over the area inside the cavity and yields
Fca v ity ⫽ ␲ b 2 B 20 .
共23兲
In order to evaluate the integral for the outside region, we
rewrite it according to
F⫽
冕
d 2 r B• 兵 B⫹“⫻ 共 “⫻B兲 其 ⫹
冖
dS•B⫻ 共 “⫻B兲 .
共24兲
The contribution from the volume integral in Eq. 共24兲 follows directly from the London equation, and we find
冕
d 2 r B• 兵 B⫹“⫻ 共 “⫻B兲 其 ⫽
2␲
B 共 r 0 ,0兲 .
␬
B 0⫽
2q
␬b2
,
F⫽ ␲ b 2 B 20 ,
as should be expected. It is easy to derive the result for a
cavity containing q flux quanta, without an external vortex.
In this case we obviously find a rotationally symmetric solution with the induction given by
K 0共 r 兲
,
K 0共 b 兲
B 共 r 兲 ⫽B 0
B 0⫽
冖
冕
F⫽
共25兲
d ␾ B 共 r, ␾ 兲 .
再
d ␾ B 2 共 r, ␾ 兲 ⫽2 ␲ 关 B 1 共 r 兲 ⫹B̃ 0 共 r 兲兴 2 ⫹
2␲q2
K 0共 b 兲
␬ 2 b K 1 共 b 兲 ⫹ 21 bK 0 共 b 兲
兺
B c2 共 b,0;b 兲 ⫽⫺
共26兲
2
n⫽0
冎
B̃ n 共 r 兲 2 .
共27兲
⳵ B̃ n 共 r 兲
⳵r
冏
⫽
r⫽b
K n共 r 0 兲
,
␬ bK 0 共 b 兲
U VC 共 r 兲 ⫽
共28兲
dS•B⫻ 共 “⫻B兲 ⫽
2␲B0
关 ␬ bB 0 K 1 共 b 兲 ⫺K 0 共 r 0 兲兴 .
␬ K 0共 b 兲
共29兲
Adding all the pieces together we obtain the final result
F⫽ ␲ b 2 B 20 ⫹
2␲B0
2␲
B 共 r 0 ,0兲 .
关 ␬ bB 0 K 1 共 b 兲 ⫺K 0 共 r 0 兲兴 ⫹
␬ K 0共 b 兲
␬
共30兲
The latter formula can be simplified further by inserting the
expressions for B 0 and B(r). After some straightforward algebra we arrive at
F⫽
2␲
关 K 0 共 r 0 兲 ⫹qK 0 共 b 兲兴 2
␬ 2 bK 0 共 b 兲 K 1 共 b 兲 ⫹ 21 bK0 共 b 兲
⫹
2␲
␬
2
K 0 共 ␬ ⫺1 兲 ⫹
2␲ c
B 共 r ,0;r 0 兲 ,
␬ 2 0
共31兲
where we have cut off the divergence of B 2 (r 0 ,0) at a distance ␬ ⫺1 . The first term in this expression differs significantly from the result given in Eq. 共10兲 in Ref. 1 if the
condition bⰆ1 is not fulfilled. Taking the limit of large b,
with q⬀b 2 , one sees that the induction and the energy take
the form
1
K 共 ␬ ⫺1 兲 ,
␬ 0
2␲
K 0 共 r 兲 2 ⫹2qK 0 共 b 兲 K 0 共 r 兲
␬ 2 bK 0 共 b 兲
K 1 共 b 兲 ⫹ 21 bK 0 共 b 兲
⫺
we arrive at
冖
共34兲
共35兲
that the energy for a vortex at a position r⫽r0 and a cavity
containing q flux quanta smoothly changes into the result for
a cavity with q⫹1 flux quanta as the vortex enters the cavity, i.e., r 0 →b.
The potential for a vortex interacting with a column containing q flux quanta is obtained directly from Eq. 共31兲 by
dropping all terms which do not involve r 0 . We find
Using the fact that
B̃ n 共 b 兲 ⫽0,
共33兲
It follows from Eq. 共31兲 and from the observation
Performing first the integration over ␾ we find
冕
1
qK 0 共 b 兲
␬ b K 1 共 b 兲 ⫹ 21 bK 0 共 b 兲
and the energy
The surface integral can be rewritten as
b ⳵
dS•B⫻ 共 “⫻B兲 ⫽⫺
2 ⳵r
共32兲
2␲
␬
2
I 共b兲
兺n Knn共 b 兲 K 2n共 r 兲
r⬎b,
共36兲
where r is the distance from the vortex to the center of the
column. The interaction potential 共36兲 is our main result and
gives the interaction of a vortex with an insulating column of
radius b containing q flux quanta, where both b and q can be
arbitrarily large. The potential U VC (R) is plotted in Fig. 1 for
a number of different values of b. It vanishes exponentially
fast at large distances, i.e., for rⰆ1.
IV. SOLUTION USING IMAGE VORTICES
If b,r 0 Ⰶ1, the problem can be solved easily using image
vortices, transcribing a problem from the textbook by Landau and Lifshits6 for a charged line in an infinite dielectric
near a cylindrical cavity with different dielectric constant, as
has first been pointed out by Buzdin and Feinberg.7,8 Assuming the magnetic induction from a vortex to decay logarithmically, we can satisfy the boundary conditions for the cavity by placing q⫹1 positive vortices in the center and a
negative vortex at the distance l⫽b 2 /r 0 from the center. The
magnetic induction is then given by
B 共 r兲 ⫽⫺
1
关共 q⫹1 兲 ln共 兩 r兩 兲 ⫹ln共 兩 r⫺r0 兩 兲 ⫺ln共 兩 r⫺l兩 兲兴 ,
␬
共37兲
where l⫽(b 2 /r 0 ,0). The energy of the system can then be
computed by integration as above and becomes particularly
INTERACTION BETWEEN A VORTEX AND A COLUMNAR . . .
PRB 62
12 411
London equation exactly using image vortices and we find
the induction
B 共 x,y 兲 ⫽He ⫺x ⫹
1
兵 K 共 冑共 x⫺d 兲 2 ⫹y 2 兲
␬ 0
⫺K 0 共 冑共 x⫹d 兲 2 ⫹y 2 兲 其 ,
共40兲
for x⬎0. The energy can again be calculated using partial
integration. The bulk term gives the contribution
Fbulk ⫽
2␲
2␲
2␲
2␲
B 共 d,0兲 ⫽
He ⫺d ⫹ 2 K 0 共 ␬ ⫺1 兲 ⫺ 2 K 0 共 2d 兲 ,
␬
␬
␬
␬
共41兲
where we again have cut off the divergence at a distance
␬ ⫺1 . The surface term is simply an integral along the y axis
and we find
冖
dS•B⫻ 共 “⫻B兲 ⫽⫺
FIG. 1. The vortex-column potential for a number of different
values of b: b⫽0.1 共solid lines兲, b⫽0.2 共dashed lines兲, and b
⫽0.5 共dotted lines兲. The upper curve corresponds to the exact solution U VC (r) and the lower curve to the potential U IV (r) obtained
using the method of image vortices in all three cases.
simple in the case of q⫽0, since the surface integral vanishes. We then find the energy
F⫽
2␲
␬2
再冉 冊
ln 1⫺
b2
r 20
冎
⫺ln共 ␬ ⫺1 兲 .
共38兲
The interaction is again given by the r 0 dependent part of Eq.
共38兲 and we finally obtain4
U IV 共 r 兲 ⫽
2␲
␬2
冉 冊
ln 1⫺
b2
r2
.
共39兲
The interaction potential 共39兲 is also shown in Fig. 1 and
compares favorably with the exact result for bⱗ0.2. In the
case of larger columns, the image vortex solution produces a
too long-ranged interaction. Note in particular that the
U IV (r) approaches zero as (b/r) 2 , whereas the exact potential U VC (r) decays exponentially for large distances.
V. BEAN-LIVINGSTON BARRIER
The main problem with the result in Ref. 1 is that it fails
to describe the limit of a large column. Sending the column
radius to infinity, one expects to obtain the interaction between a vortex and a flat insulating wall, i.e., the BeanLivingston barrier. We show below that this is indeed the
case for the interaction in Eq. 共36兲. For the sake of completeness, however, we first derive the Bean-Livingston barrier
using methods similar to those above. Consider an interface
to an insulator, which we take to coincide with the y axis and
a vortex at a distance d from the wall at position r⫽(d,0).
The symmetry of the problem makes it possible to solve the
冕
dy B ⳵ x B
冕
⫽
2Hd
␬
⫽
2 ␲ H ⫺d
e .
␬
dy
K 1 共 冑d 2 ⫹y 2 兲
冑d 2 ⫹y 2
共42兲
The integral was computed using the substitution y
⫽d sinh(x) and Eq. 共6.6648兲 in Ref. 11. Adding all the parts
we find energy
FBL ⫽
4␲
2␲
2␲
He ⫺d ⫹ 2 K 0 共 ␬ ⫺1 兲 ⫺ 2 K 0 共 2d 兲 .
␬
␬
␬
共43兲
Dropping irrelevant terms, we arrive at the well-known
Bean-Livingston barrier,
U BL 共 d 兲 ⫽
4␲
2␲
He ⫺d ⫺ 2 K 0 共 2d 兲 ,
␬
␬
共44兲
which we have normalized so that it vanishes for large d.
This well-known result can be found in many textbooks on
superconductivity 共see, e.g., Ref. 12兲.
VI. INFINITELY LARGE COLUMN RADIUS
We now turn to the case of an infinitely large cavity and
show that this reproduces the Bean-Livingston result above.
We want to have a finite induction in the cavity and we
therefore need a large number of flux quanta, q⬀b 2 . Furthermore, we want the distance of the vortex from the edge of
the cavity to be finite and write r 0 ⫽b⫹d. Inserting this into
Eq. 共36兲 and discarding irrelevant terms we find
U VC 共 b⫹d 兲 ⬇
8 ␲ q K 0 共 b⫹d 兲 2 ␲ c
⫹
B 共 b⫹d,0兲 . 共45兲
␬ 2
␬ 2b 2 K 0共 b 兲
Using the fact that we have B 0 ⬇2q/ ␬ b 2 , we can write this
as
H. NORDBORG AND V. M. VINOKUR
12 412
PRB 62
We did not find a simple proof of Eq. 共48兲, but the result can
be understood in the following manner: We begin by observing that
K 0 共 2d 兲 ⫽
兺n I n共 b 兲 K n共 b⫹2d 兲
共49兲
for any b. It then follows that
K 0 共 2d 兲 ⫹ ␬ B c2 共 b⫹d,0兲
⫽
兺n I n共 b 兲 K n共 b⫹2d 兲
再
1⫺
冎
K n 共 b⫹d 兲 2
.
K n 共 b 兲 K n 共 b⫹2d 兲
共50兲
Inserting the lowest order asymptotic expansion we find
1⫺
FIG. 2. Comparison of the Bean-Livingston barrier 共solid line兲
with the vortex-column interaction for large columns (b
⫽5,10,20). We always have q⫽b 2 flux quanta in the column 共in
dimensionless quantities兲, producing a magnetic induction of B
⬇q⌽ 0 / ␲ b 2 in the limit of large columns. The convergence to the
BL result is slow, but already the smallest column shows the correct
characteristic shape.
U VC 共 b⫹d 兲 ⬇
4 ␲ K 0 共 b⫹d 兲 2 ␲ c
B
⫹
B 共 b⫹d,0兲 .
␬ 0 K 0共 b 兲
␬ 2
共46兲
For the first term we have the approximation
4 ␲ K 0 共 b⫹d 兲 4 ␲
B
⬇
B e ⫺d
␬ 0 K 0共 b 兲
␬ 0
冑
b
4␲
⬇
B e ⫺d ,
b⫹d ␬ 0
lim ␬ B c2 共 b⫹d,0兲 ⫽⫺K 0 共 2d 兲 .
共48兲
b→⬁
*Present address: ABB Corporate Research Ltd., 5405 BadenDättwil, Switzerland.
1
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5
V. Metlushko, U. Welp, G.W. Crabtree, R. Osgood, S.D. Bader,
L.E. DeLong, Zhao Zhang, S.R.J. Brueck, B. Ilic, K. Chung, and
P.J. Hesketh, Phys. Rev. B 60, R12 585 共1999兲.
2
,
共51兲
which shows that the right-hand side of Eq. 共50兲 vanishes in
the limit b→⬁, a result which we have also verified numerically. In Fig. 2 we plot the Bean-Livingston barrier 共43兲
compared to the column-vortex interaction 共36兲 for a number
of different column sizes. The convergence is fairly slow, but
already the column of the size b⫽5 shows the typical BeanLivingston shape.
VII. CONCLUSION
We have derived the interaction between a vortex and an
arbitrarily large cylindrical cavity in a type-II superconductor
using the London approximation. In the limit of an infinitely
large cavity, the interaction reduces to the well-known BeanLivingston result. Our work extends an earlier work on the
same subject.
ACKNOWLEDGMENTS
共47兲
in agreement with Eq. 共44兲. It thus remains to show that
冉冊
1 d
K n 共 b⫹d 兲 2
⬇
K n 共 b 兲 K n 共 b⫹2d 兲 2 b
The authors want to thank A. E. Koshelev for valuable
discussion. This work was supported at Argonne by the US
Department of Energy BES–Materials Sciences under Contract No. W-31-109-ENG-38.
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